In the analytical
geometry field, the “curves family” term is well-known,
representing a set of continuous functions on a common interval,
which are different one from another as a symbolic dependence
relation only through the value of a parameter. For example, the
family of the straight-lines which cross through the origin have a
generic relation *y = mx*, where *m* (angular coefficient)
is the parameter; in particular, in case of the straight-lines which
have an angular direction ranging within the interval
against
the axis **X, ***m *may take any value between -1 and +1. In
this case, the family of curves does not have asymptotes, the
parameter *m* being also able to take the extreme value (the
interval’s boundaries). The situation is different when there
is a certain numerical value which allows the parameter to get close
to it as much as possible, but if this value would be reached, it
would qualitatively alter the object model (distribution type), this
new object not being included into the family any longer. For making
an illustration of this situation, we shall take into consideration
the figure X.3.2.2.1 with the hyperbola family from the first
quadrant *xy=C*, where C is a positive numerical constant
(parameter), with a value specified for a certain hyperbola from the
family (a certain member of the family).

*Fig. X.3.2.2.1.1*

It is obvious that
the axes of reference are the asymptotes of the hyperbola family for
the parameter *C* tending to zero, but if we would accept that
the hyperbola and their asymptotes belong to the same family
(therefore, they have the same model, the same symbolic distribution
relation), we would reach to the absurd conclusion that even the axes
of reference are a hyperbola (but a edited one, isn’t it? (#)).
The same situation is in case of the relations X.3.2.2.3 and
X.3.2.2.4, which define a group of ratios of some symmetrical finite
differences (left/right) of a function against a reference value, a
group whose parameter is ***x*.
The __directional asymptote__ of this family is the direction of
the tangent in the point *P*, object towards which the two
variations tend to, but it cannot substitute them because they are
qualitatively different objects.

Now, let us
interpret the above-mentioned data by using the terminology specific
to this paper, which is introduced in chapters 2…9. Thus, in
case of the “families”, it is clear that we are dealing
with classes of abstract objects, a *class* being an abstract
object which has the *common component* of the models from a set
of objects (members, class instances). In case of the straight-lines
which cross through the origin or as regards the aforementioned
hyperbolas, the common component of all the class members is the
general symbolic relation (*y=mx* for the straight-lines and
*xy=C* for the hyperbolas), each instance of those particular
classes being differentiated only by means of the parameter’s
numerical value, that is a value which represents, in terms of the
objectual philosophy, the *differential* (specific)* component
*of a member belonging to the class of curves (against the other
members). While in case of the straight-line classes which cross
through the origin, the definition relation of the class do not
change regardless of the parameter value, as for the hyperbolas, an
interesting aspect occurs when the parameter tends to zero, namely,
as long as this parameter *(C)* is different from zero, this
means that the hyperbolic dependence relation between *y* and *x*
__still exists__; when the parameter vanishes, the dependence
relation vanishes as well, and the two amounts become __independent__
(axes of reference), as we have pointed out in chapter 2 about the
independence concept regarding the variables. It is very clear that
in this case, the limit objects of the class (asymptotes) and the
normal objects within one class are qualitatively different as a
class model, while the class members are different one from another
only by means of the specific component values.

Copyright © 2006-2011 Aurel Rusu. All rights reserved.